Directed Distance in Coordinate Geometry: Sign Convention

[batballbat]

in coordinate geometry i am having problem with the sign convention of directed distances. Let P1 and P2 be arbitrary points on the graph. Then what is the sign convention for P1P2 to be positive or negative. I know that if P1P2 is parallel to x-axis or y-axis then the normal convention for positive and negative direction (right=positive...up=positive). But what happens when P1P2 is not parallel to x-axis or y axis. What is the convention to determine whether P1P2 is positive or negative. Please exhaust all the possible cases.

 

[cepheid]

Distances are always positive, so the distance between P1 and P2 is positive.

As for the displacement...displacement is a vector. A signed scalar quantity is not sufficient for describing a displacement in general. In the special case where the displacement lies entirely along one of your chosen coordinate axes, you can do it (because it reduces to a 1D situation). But the short answer to your question is, I think, that the displacement doesn't have a "sign" because it can't be described using a single number. It is a vector quantity (there are infinitely many directions in which it can point, as opposed to just two).

EDIT (to elaborate on this further): you need at least two numbers to describe a vector (if you're in a 2D space, that is). These numbers could be x and y components (in which case either component could be either positive or negative). Alternatively, the two numbers could be the magnitude of the displacement and a bearing/direction, the latter of which is just an angle measured relative to some chosen reference direction.

 

[batballbat]

in my book P1 is above P2 and it says that P1P2 is negative

 

[batballbat]

quoted from my book

If the segment is parallel to the x-axis, we say that its
positive sense is that of the positive direction of the x-axis.
If the segment is not parallel to the x-axis, we make the convention
that upward along the segment is the positive sense on
the segment.

 

[cepheid]

in my book P1 is above P2 and it says that P1P2 is negative

So your book assigns a sign to the "directed" segment that is basically the sign of the y-component of the vector between the two points in the segment. Fine.

Obviously there are four possibilities:

1. x-component is positive, y-component is positive
2. x-component is positive, y-component is negative
3. x-component is negative, y-component is positive
4. x-component is negative, y-component is negative

Your book would call 2 and 4 'negative' directed segments. But I think you can probably see the advantage of just working with vectors, rather than doing what they do. They assign an arbitrary convention for directions "along" a segment, but it's not really necessary.

 

[batballbat]

thanks